涨潮的海

A phrase with a source

The chapter-title metaphor is not a last utterance. It is not a description of Grothendieck’s death, and it should not be made into one. Colin McLarty translates and analyzes the title and image of a note in Récoltes et Semailles. He locates the passage at pages 552 to 555 in the version he consulted and identifies his translations as his own. In his rendering, a resistant mathematical object is not split by force; a widening body of theory advances until the object is surrounded and the theorem becomes a consequence within a larger structure. 11McLarty, “The Rising Sea: Grothendieck on Simplicity and Generality,” 1–2.

That chain of attribution matters at the end of a biography. The French wording belongs to Grothendieck. The English wording belongs to a translator. The exact primary passage still needs checking in an authorized edition before it can stand alone as an epigraph. The metaphor has been widely repeated, but repetition is not verification. This account therefore keeps the image as the chapter’s title and does not use the quotation as an epigraph. 22McLarty, “The Rising Sea: Grothendieck on Simplicity and Generality,” 1–2.

The caution is not pedantry placed against beauty. It is part of the meaning. Grothendieck took naming seriously because a name organizes attention. McLarty’s selected rendering suggests patience, enveloping generality, and an almost imperceptible change of level; another plausible rendering would stress recurrence and shore. A translation decides which associations enter English. To mark that decision is to respect both languages.

The image can now do limited work. It can describe one mathematical style documented by a scholar who connected it to schemes, abelian categories, topoi, and the Weil-conjecture program. It can illuminate the way a broad language turns a special obstruction into a natural case. It cannot explain every proof, erase calculations, or assign all collective work to the person who described the style. It cannot serve as a mystical law of history.

What rose after the life

After November 2014, the level of public access changed. The University of Montpellier inventoried, conserved, and digitized papers already deposited there. In May 2017 it placed approximately 18,000 of approximately 28,000 pages online, covering principally mathematical work from 1949 to 1991. In late September 2023 the BnF announced the entry of another corpus, combining a bequest with a gift from Grothendieck’s children and including forty-one volumes of Réflexions sur la Vie et le Cosmos. 33Université de Montpellier, “28 000 pages d’Alexandre Grothendieck”; Institut des Hautes Études Scientifiques, “Some of Grothendieck’s Archives Published Online by the University of Montpellier”; Réseau national des bibliothèques de mathématiques, “Archives mathématiques d’Alexandre Grothendieck: 18 000 documents en libre accès”.

These acts raised visibility. They did not raise every page into publication or every thought into public property. Montpellier’s portal provided scans under institutional terms and within an access selection. The BnF provided custody, description, and research conditions for a distinct body. Rights in writings, scans, photographs, editions, translations, and correspondence remained layered. Any statement about present access belongs to a date, because interfaces, catalogs, permissions, and restrictions can change. 44Bibliothèque nationale de France, “Reproduire une œuvre protégée au titre du droit d’auteur”; Bibliothèque nationale de France, “Reproduction des documents”; Code de la propriété intellectuelle, art. L121-1.

Public interpretation also widened. Gallimard’s two-volume Récoltes et Semailles appeared in 2022, followed by critical discussion about its editorial scope. La Clef des Songes appeared from Éditions du Sandre in 2024. A January 2025 BnF-SMF study day brought together activism, teaching, mathematics, film, autobiography, and late spiritual writing. A twelve-panel exhibition based substantially on Montpellier material traveled from Toulouse to Paris venues, including ENS-PSL in 2026. 55Grothendieck, Récoltes et semailles; Institut des Hautes Études Scientifiques, “Publication of *Récoltes Et Semailles* by Alexander Grothendieck”; Centre for Grothendieckian Studies, “*La Clef Des Songes* Published by Éditions Du Sandre”.

The widening was not smooth agreement. Grothendieck’s 2010 declaration against publication remained part of the record. French moral and disclosure rights supplied legal structures that could not be replaced by scholarly enthusiasm. Critics disputed accounts of credit and editorial choices. Families and institutions held responsibilities that mathematicians did not. That friction is not debris around a pure legacy. It is the form in which the legacy exists. 66Grothendieck, “Déclaration d’intention de non-publication,” 1; Bibliothèque nationale de France, “Reproduire une œuvre protégée au titre du droit d’auteur”; Bibliothèque nationale de France, “Reproduction des documents”.

Mathematics does not wait for a complete archive

While manuscripts moved into catalogs, mathematics continued through routes that were already open. Schemes and base change remained the grammar of algebraic geometry. Sites and sheaves continued to define useful notions of locality. Stacks preserved automorphisms and descent in moduli. Derived geometry retained higher information that ordinary intersections lose. Motives connected cohomological visions with rigorous theories and unresolved conjectures. Dessins and anabelian programs kept arithmetic questions attached to surfaces, graphs, and fundamental groups. 77The Stacks Project Authors, The Stacks Project, 1; The Stacks Project Authors, The Stacks Project; Grothendieck, “Esquisse d’un Programme,” 7–48; Grothendieck, “Esquisse d’un Programme,” 003.

The paths did not require the claim that Grothendieck had completed them. The Stacks Project turned foundations into a maintained collaborative reference. Toën, Vezzosi, Lurie, and many others built forms of derived and higher geometry. Voevodsky and Morel developed motivic machinery and proved theorems. Scholze’s diamonds and $v$-stacks carried sites, descent, étale cohomology, and six operations into a new $p$-adic setting. Caramello made classifying topoi into systematic bridges between theories. Each path has its own authors and its own technical necessity. 88Toën, “Derived Algebraic Geometry”; Caramello, Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic “Bridges”, chs. 1–4.

This is a more exact form of influence than the statement that Grothendieck “is everywhere.” Ubiquity cannot be checked. A definition with a tag can be checked. A paper’s citations can be checked. A later author’s account of inspiration can be checked. A theorem’s hypotheses can be compared with an earlier program. The point is not to reduce intellectual history to citation counts; it is to replace atmospheric ownership with relations that readers can examine.

The metaphor helps only if its limit remains visible. A theory can make a theorem appear natural, but the theory itself has to be built. Definitions are tested against examples. Coherence is proved. Counterexamples force hypotheses. Expositions are rewritten. Students and collaborators find errors. Later generations decide which generality is fruitful and which is inert. Conceptual change is labor, not weather.

The names beneath “Grothendieckian”

An adjective can preserve a source and conceal a crowd. “Grothendieckian” may refer to relative thinking, categorical formulation, ambitious generality, particular topologies, a style of exposition, or a moral stance toward mathematics. Unless the noun is supplied, the adjective can make disagreement look like discipleship and collective work look like emanation.

The earlier Parts of this biography have named collaborators where the documentary record allows: Jean Dieudonné in the writing of EGA; speakers and editors across SGA; Jean-Pierre Serre in the exchange of problems and ideas; students who turned programs into theses and theories. The later story adds different forms of collaboration. Editors transcribe typescripts. Maintainers accept corrections. Archivists create cotes. Designers construct exhibitions. Rights holders authorize or refuse uses. These acts are not mathematically equivalent, but all shape what can be inherited.

Credit must therefore survive the end. If a result is Deligne’s proof, the sentence says so while explaining the Grothendieckian framework it used. If a theorem is Belyi’s, the sentence does not absorb it into the later fascination with dessins. If a theory is Scholze’s, structural ancestry does not alter authorship. If an electronic edition was prepared by named editors, “Grothendieck’s text online” is incomplete description.

The rule also protects the subject from exaggerated praise. A claim that one mathematician created the language of all modern geometry sets up an easy correction and makes real achievements harder to see. Grothendieck’s influence does not need maximal borders. Schemes, topoi, cohomological formalisms, motives, and late programs are large enough when described accurately.

An archive is not a theory

Archival scale and theoretical breadth can be confused. A mass of pages may suggest that sufficient reading will disclose a total system. An archive instead consists of finite material, unevenly preserved and institutionally arranged. It contains starts, repetitions, corrections, and blank backs. A theory has different relations. The distinction matters because archival work advances through boundaries: series, boxes, volumes, cotes, leaves, dates, and access rules.

Montpellier’s scale is meaningful only through those boundaries. Approximately 18,000 online pages do not form one work. They include different genres over four decades. The BnF’s forty-one late volumes have an authorial organization significant enough to preserve, but even a meticulous sequence does not dictate how biography should excerpt it. The page before and the page after remain the first protection against a seductive line.

The traveling exhibition demonstrates selection openly. Twelve panels cannot claim totality. Maronne and Toën chose the mathematical activity from 1973 to 1991 and explained that choice; Philomène Longchamp made a graphic form for it. The exhibition’s mobility gives the archive new audiences without moving the originals. Its limitations—a specialist emphasis, finite panels, permissions required for reuse—are conditions of intelligibility rather than defects. 99Maronne and Toën, “Préface”; “Grothendieck mathématicien: Le temps des réflexions 1973–1991 (exposition)”; Bibliothèque nationale de France and Société mathématique de France, “Alexandre Grothendieck, mathématicien et militant”.

A biography is another exhibition, built from sentences rather than walls. It selects one route through childhood, institutions, mathematics, activism, withdrawal, and afterlife. The only defense against the route’s inevitability is visible sourcing and room for alternatives. A reader should be able to leave the narrative and find the record.

The person does not dissolve

In mathematics, conceptual enlargement can make a resistant theorem a consequence of theory. In biography, such dissolution is a danger. A person cannot be made the inevitable consequence of adversity, political conviction, mathematical style, or spiritual desire. General context can explain possibilities and constraints; it cannot remove singularity or responsibility.

Nor should the dead person dissolve into a brand. Grothendieck’s name now identifies institutes, study days, exhibitions, online collections, books, groups, and research programs. Naming can sustain memory and attract resources. It can also smooth contradiction into cultural capital. The biography should ask what each use does: preserve papers, teach mathematics, authorize an edition, claim a lineage, sell a product, or gather a community.

The 2010 declaration makes the resistance personal. Even if later legal processes permit particular acts, the author’s objection remains evidence about his relation to publication. The bequest to the BnF makes the relation more complex rather than cancelling it. He could oppose dissemination in one form and provide for preservation in another. Without the testamentary documents and work-specific context, the biography should not force those acts into either contradiction or reconciliation. 1010Grothendieck, “Déclaration d’intention de non-publication,” 1; Code de la propriété intellectuelle, art. L121-1; Code de la propriété intellectuelle, art. L121-2.

The late spiritual manuscripts require the same refusal to dissolve. They are not simply the ruins of a mathematician, nor automatically the crown of his thought. They are works with dates, internal structures, revisions, and claims that need the appropriate kinds of reading. A diagram in Structures de la Psyché may reveal how Grothendieck carried structural habits into another domain; it does not convert psychology into proven mathematics. Thousands of pages on evil show sustained preoccupation; page count does not settle truth.

The horizon in August 2026

At the time of writing, several statements can be made securely. Montpellier had launched substantial online access in 2017 and continued to be cited by institutional pages in 2025 and 2026. The BnF had accessioned and cataloged the late corpus under NAF 29093, published a press description, and convened a study day. The mobile mathematical exhibition had reached ENS-PSL for a showing ending 8 July 2026. Gallimard’s Récoltes et Semailles and Éditions du Sandre’s La Clef des Songes existed as commercial editions. 1111Hartemann et al., “Les mille et une pages de Grothendieck”; Bibliothèque nationale de France and Société mathématique de France, “Alexandre Grothendieck, mathématicien et militant”; École normale supérieure – PSL, “Exposition: Grothendieck, le temps des réflexions, 1973–1991,” 8, 2026.

Several stronger statements cannot be made from the sources gathered here. This account cannot say that every Montpellier page remains reachable at every moment, that the BnF late manuscripts are digitized for public web access, that all posthumous editions share one authorization history, that every image visible on an archival website may be reproduced, or that the record is closed. These are not rhetorical caveats. They are boundaries in the evidence.

The present tense should therefore carry a timestamp. “The BnF holds” refers to a catalog checked at the time of writing. “The exhibition travels” refers to a project with documented venues, not a promise of future bookings. “The archive is online” refers to an institutional service whose current item-level operation must be rechecked. “The rights are protected” states a general legal framework, not the identity or decision of every rights holder.

Future research may resolve some issues. A provenance note may reconcile 1990 with 1990 and 1995. A publisher may document permissions. A catalog may expose finer component dates. A mathematician may find that a notebook contains a useful lemma. Another may show that an apparent anticipation is incomplete. The book should be constructed so that such corrections can enter source cards without requiring the story to defend its pride.

What remains to be done

The actions are ordinary and exact. Archivists preserve the physical order and describe changes. Editors collate versions and disclose decisions. Mathematicians check arguments and credit both programs and proofs. Historians compare archives rather than allowing the largest to dominate. Translators identify their choices. Image editors secure item-level provenance and permission. Lawyers review formal-publication rights. Readers retain the right to disagree.

This book follows the same route. Direct quotations carry exact sources. Mathematical explanations carry definitions, sources, and stated limits of analogy. Images carry their creator, licence and source.

This procedure may seem unlike an ending. It is the ending appropriate to material still being described. A biography can close its narrative without claiming that cataloging, interpretation, or mathematical development has closed. The distinction keeps the present from masquerading as eternity.

A final return to simplicity

Grothendieck’s mathematical simplicity was not always brevity. It often meant changing the setting until a difficult statement belonged to a natural pattern. The resulting exposition could be vast because it built the setting carefully. The paradox helps explain both the attraction and the burden of his work. A simple relation may require hundreds of pages of foundations; a short proof may rest on a communal architecture.

The life resists the same word. Nothing simple can collect Grothendieck’s documented roles as displaced student, mathematician, teacher, activist, polemicist, spiritual writer, father, correspondent, and withdrawn older man without loss. A biography seeking one essence would make simplicity by deletion.

A more restrained simplicity appears in acts. Grothendieck wrote. Other mathematicians read, argued, extended, or refused. Malgoire preserved papers. Montpellier inventoried and scanned. The children and the BnF completed an archival transfer. Editors made books. Curators selected panels. Institutions offered rooms and websites. Each verb can be sourced; none explains the whole person.

At this final scale, the metaphor is useful only after losing any triumphal force. Knowledge does not accumulate evenly. Some pages remain restricted, illegible, or misunderstood. Some mathematical programs flourish; others recede. Ethical clarity can lag behind technical access. Change also erodes, obscures, and separates. No figurative pattern should promise that time resolves every dispute.

The last sentence’s limit

There is no sourced last scene to offer. No manuscript line has the right to stand for all the others. No theorem closes the life, and no archive restores its author. The ending can identify only a relation among absences and continuities.

Grothendieck is absent from the arguments now conducted in his name. The papers remain present under the care of institutions and successors. The mathematics remains active only because other people make new mathematics with, beyond, and sometimes against the structures and programs associated with him. The biography remains accountable to all three facts.

So the final image is not a solitary figure beneath an advancing natural force. That would be an invented scene. It is a set of documented practices, quieter than legend: a leaf receives a cote; a conjecture receives a hypothesis; a quotation receives a page; a photograph receives a credit; a claim receives a correction. Around resistant stories, context accumulates slowly. It does not dissolve the person. It makes the limits of what can be said visible—and leaves the work open to the next reader.

Memorials that do different work

An obituary, archive, edition, conference, exhibition, institute, and theorem name are all called memorials in ordinary speech. They do different work. An obituary compresses a life at the moment of loss. An archive preserves evidence beyond any one summary. An edition makes a selected text readable. A conference creates a temporary community of interpretation. An exhibition gives selected documents spatial form. An eponym attaches a name to daily technical use.

Confusing these forms produces misplaced expectations. An obituary cannot supply a critical apparatus. An archive does not tell visitors which pages matter. A conference program does not certify agreement among speakers. An edition can be meticulous without being complete in every possible sense. An eponym can preserve memory while obscuring collaborators.

The post-2014 record contains each form. CNRS and mathematical institutions marked the death. Montpellier created digital access. Gallimard and Éditions du Sandre issued books. The BnF accepted manuscripts and convened a study day. Maronne, Toën, and Longchamp made a mobile exhibition. Mathematical communities continued to use schemes, topoi, motives, stacks, and programs bearing or evoking Grothendieck’s name. 1212Université de Montpellier, “28 000 pages d’Alexandre Grothendieck”; Bibliothèque nationale de France, “Les manuscrits du mathématicien Alexandre Grothendieck entrent à la BnF,” 1–2; Bibliothèque nationale de France and Société mathématique de France, “Alexandre Grothendieck, mathématicien et militant”.

No form should be asked to close the others. A national acquisition does not replace a family history. A new edition does not replace manuscript study. A mathematical theorem does not resolve the ethics of publication. The plural memorial landscape is more accurate than a monument imagined at its center.

This plurality also distributes power. A publisher determines an edition’s form; a repository determines access; curators determine panels; mathematicians determine whether an argument works; readers determine what questions persist. Accountability improves when each actor’s mandate is limited and named.

The environmental resonance and its restraint

For a reader who knows Grothendieck’s ecological activism, the chapter-title wording can acquire a contemporary environmental meaning. McLarty notes a semantic range extending from periodic motion to long-term level change. That range is the translator’s observation. The mathematical passage describes a style of approach; it is not a documented prediction about climate. 1313McLarty, “The Rising Sea: Grothendieck on Simplicity and Generality,” 1–2.

The resonance should not be exploited into a final emblem that fuses mathematics and activism without evidence. Grothendieck did warn about technological and ecological danger through Survivre et Vivre. He also used natural and growth images to describe mathematical creation. A biography may place those languages near each other. It cannot claim that the mathematical image secretly encoded environmental politics unless the text makes that connection.

Restraint is especially necessary because climate change gives the wording an urgency different from its 1980s literary context. The modern reader supplies that urgency. Identifying the reader’s association can be illuminating; attributing it backward would be anachronism.

The ethical lesson lies elsewhere. Metaphors travel beyond authorial control, just as manuscripts and mathematical concepts do. Later contexts add meaning. Good interpretation records the new context and keeps the source context available. A metaphor can remain fertile without being made prophetic.

Time scales

The afterlife of mathematics operates on several clocks. A preprint can alter a field in weeks. A seminar volume can remain standard for decades. A conjectural program may wait for tools invented by another generation. An archive can take years to process and a century to exhaust. Copyright follows statutory time. Privacy can depend on living people. Paper deteriorates continuously.

Grothendieck himself imagined that work might interest someone in ten years or a hundred, according to a passage quoted by the BnF from Récoltes et Semailles. The BnF article gives the wording but not the exact primary page, so this account does not quote it. The institutional choice to quote it nevertheless reveals how the library frames long stewardship: immediate comprehension is not required for preservation to have purpose. 1414Hartemann et al., “Les mille et une pages de Grothendieck”.

A biography has a shorter production clock and must resist pretending to represent the end of longer processes. It freezes sources, records open issues, and becomes one dated intervention. Its value will depend partly on how easily later readers can correct it.

The different clocks explain why uncertainty is not failure. A repository can preserve a volume before specialists know its significance. A rights holder can delay reproduction while legal questions are checked. A mathematician can publish a partial result. A historian can state that two dates conflict. Responsible work matches the claim to the available time.

Urgency remains real where material is fragile or witnesses are aging. Oral history, conservation, and provenance research can be time-sensitive. Urgency justifies resources and prompt action; it does not justify fabricated certainty. The faster the intervention, the more important its documentation.

The reader as successor

A successor in law holds authority of a kind a reader does not. A successor in mathematics proves or teaches something new. A successor in reading has a smaller role: to receive a text with enough context to question it. This book can prepare that role without appointing its reader guardian of Grothendieck’s intention.

The reader should know where a direct quotation begins and ends. The reader should be able to distinguish an archive count from a work count, an institutional claim from a theorem, a tentative epigraph from a verified quotation, and a current access link from a perpetual promise. These distinctions are not backstage production details. They shape trust.

The reader also inherits disagreement. Schapira’s criticism of Récoltes et Semailles need not be resolved into admiration or rejection. Grothendieck’s accusations can be tested without denying the ethical questions he raised. His spiritual writings can be studied without diagnosis or assent. His withdrawal can be described without making solitude sacred.

Mathematical readers carry another obligation. Familiarity can hide history: a fiber product or site looks inevitable after training. Returning to the problem that made the concept necessary restores contingency. Historical readers carry the converse obligation not to treat difficult mathematics as interchangeable symbolism. Each group needs the other’s checks.

The next reader may be a cataloger who corrects a date, a student following a permanent tag, a relative protecting a third party, a translator comparing a phrase, or a mathematician reconstructing a proof. “Legacy” becomes less abstract when those roles are imagined concretely.

Failure modes of the grand ending

One grand ending would proclaim immortality: Grothendieck dies, but his ideas live forever. The evidence cannot support “forever,” and the formula erases the maintenance by which ideas remain readable. Mathematical languages can fall from use; files can break; institutions can lose funds; attributions can change. Continuity is achieved, not guaranteed.

A second ending would make the archive a resurrection. The BnF volumes and Montpellier scans provide extraordinary access to writing, but no page can answer a new question from the historian. Texts preserve traces of a person’s activity, not presence. Calling them a return of the man would confuse mourning with evidence.

A third ending would stage reconciliation. Posthumous editions and conferences bring Grothendieck’s words back into mathematical institutions, but they do not establish that he would have approved, that former disputes are healed, or that family relations are repaired. The 2010 declaration keeps easy reconciliation unavailable.

A fourth would turn him into a prophet of all later mathematics and ecology. Specific programs did anticipate fertile directions; specific warnings belong to environmental history. Prophecy is a retrospective category that can flatten the many authors who converted questions into results and the many activists who shared concerns.

A fifth would punish the grand endings by ending in debunking. Correcting exaggeration does not require reducing Grothendieck’s achievements or the scale of his influence. Precision is not disenchantment. Schemes, topoi, cohomological architecture, motives, and late programs remain consequential when proper names and limits are restored.

The restrained ending is harder because it leaves several emotions unresolved. Admiration, discomfort, grief, anger, curiosity, and intellectual pleasure can coexist. The book need not choose one as the reader’s verdict.

The material future

The physical manuscripts will require conservation after current readers are gone. Ink, paper, bindings, storage environments, and handling impose conditions independent of intellectual fame. Digital surrogates reduce handling and introduce their own dependencies on formats, storage, metadata, and institutional continuity.

An illustrated biography participates in that material future on a small scale. If it reproduces a leaf accurately, with cote, color control, permission, caption, and alt text, it creates another accountable route to the object. If it uses an unattributed low-resolution image as texture, it spreads uncertainty. Production ethics is historical method embodied in a file.

Rendered DOCX and PDF products can then change design without severing evidence. Hashes can identify final image files. A later correction can locate every affected occurrence.

The book’s own preservation should not be confused with the archive’s. It selects and interprets; it cannot serve as a backup for restricted collections. Its sources may link to services that change. A bibliography and archival guide should supply enough institutional information for readers to begin again.

This material modesty answers the desire for permanence better than monumental language. A well-made book remains finite, versioned, and corrigible. Its care can honor a legacy without claiming to contain it.

A final mathematical caution

The method can encourage generality for its own sake. Some of Grothendieck’s generalizations proved productive because they organized concrete problems and enabled new operations. Other abstractions can obscure examples, impose costs, or solve no pressing problem. The metaphor describes one documented mathematical style; it is neither a universal method nor a personal verdict.

Mathematical cultures need several styles. Serre’s concision, computational experiment, classification, explicit example, probabilistic method, and conceptual architecture answer different questions. Grothendieck’s own account contrasted approaches vividly, but a history should not rank all work by resemblance to his.

This pluralism is part of the legacy. A scheme can be studied through categorical properties and explicit equations. A dessin can invite a childlike image and difficult computation. Derived geometry can clarify deformation while requiring formidable machinery. Generality and specificity correct one another.

The final chapter therefore does not ask modern mathematics to become Grothendieckian. It asks that the routes from his work be named accurately and that later work be credited on its own terms. Influence is strongest when it permits difference.

Open, not unfinished

An unfinished book lacks required work. An open ending has completed its argument without pretending that its subject has stopped changing. This Part has followed the papers into institutions, the scans into an exhibition, concepts into later mathematics, manuscripts into contested publication, and evidence into the limits of biography.

The narrative argument is complete. After a life, custody changes; interpretation multiplies; mathematics continues through other authors; ethical authority remains divided; biography reaches a boundary. No final status of archive or copyright is needed to state those relations.

What stays open is correction. A new theorem can extend a path. A successor can clarify permission. A reader can reject the book’s interpretation while recovering the evidence. This is not the immortality of a name. It is the ordinary durability of work that other people remain able to examine.

Coordinates for the next account

A later historian should be able to begin from coordinates rather than this book’s authority. The Montpellier route begins with the university’s deposit and digitization records, the archive portal, and item-level cotes. The Paris route begins with NAF 29093, the BnF accession notice, and its developing component descriptions. Publication history begins with dated editions, contracts where available, and the 2010 declaration. Mathematical history begins with exact texts and later papers, not with the label “legacy.”

Those coordinates will change in resolution. A catalog may divide one component into several. New correspondence may establish circulation. A critical edition may replace a provisional transcription. Legal status may be clarified for a particular work. The later account should cite the change rather than accuse an earlier, properly qualified account of failing to know the future.

The same principle governs disagreement with this biography. A critic should be able to identify the paragraph, source and inference under dispute. If the source was misread, the claim can be corrected. If the evidence permits two interpretations, the alternative can stand beside this one. Traceability turns criticism into cumulative work.

The coordinates include absences. No item-level permission was obtained for this account. No legal opinion was issued. The full French primary passage was not verified in an authorized edition. No current global uptime claim was made for the Montpellier portal.

There is no shame in an account that marks its limits. The shame would lie in removing the marks to make inheritance look simple. A book about Grothendieck’s architectures should keep its own supports visible enough to inspect.

Those supports include the negative statements on which this ending depends. The archive is not complete access to a mind. Mathematical influence is not ownership. Custody is not copyright. Publication is not proof of every permission. Quotation is not context. A metaphor is not a cause. Each negation prevents one seductive substitution.

What remains after the substitutions are refused is substantial. There are preserved leaves, dated programs, published theories, corrected references, living research communities, and institutions that have accepted long obligations. There are also explicit refusals, disputed memories, unopened questions, and people whose interests do not merge into public curiosity.

The ending can hold both sets without balancing them into equivalence. Preservation does not cancel refusal; refusal does not make preservation meaningless. Later mathematics does not vindicate every late claim; error does not erase earlier mathematics. The book’s restraint lies in keeping relations visible where a verdict would be easier.

The next reader does not inherit an obligation to admire. The inheritance is access to enough evidence to understand why admiration, criticism, and uncertainty arose. If this book has done its work, disagreement can begin from a shared page rather than a borrowed legend, and correction can strengthen memory rather than threaten it. The final responsibility belongs not to memory in the abstract, but to people who preserve, question, prove, translate, permit, refuse, design, teach, and read now and after this book closes.

Source limits remain visible, and the title names only a method, not a destiny. That restraint keeps posthumous attention from becoming claims of possession. Future readers can then challenge the account without first undoing a myth that the evidence never warranted at any stage.